trouble with Quicktime mov files

Hi everyone. I am working out of FCP 6.06. I need to make a 10 minute compilation of a bunch of my video projects for a scholarship application. I imported into the FCP project the quicktime mov files of the original films instead of the raw clip sequences. also, i have a few graphic Motion files to mark transitions between movie excerpts.

because i did not build all of these projects on this computer, to
track down original movie files and clips might be impossible or reconnecting all the media would probably take even longer--not to mention leave me with a very messy project.

so I am really hoping there is a way to solve the problem I am having, which is this:

When I cut the quicktime MOV files into excerpts on independent sequences, they look and even playback just fine. So I made a "rough cut" sequence and copy/pasted each little excerpt into the new sequence and strung them together. I rendered everything overnight. On the canvas, everything looks okay, but in the viewer, all of the files are much smaller in this new "rough cut" sequence then they are in their original sequence. And of course, there is the problem that rendering and then compressing into a new QT rough cut are both taking an enormous amount of time for my ten minute project.

my questions are:

* How do I maintain clip integrity from one sequence to another? Or how do I change the clip attributes so they have the same attributes and do not shrink/distort in the new sequence?
* should I be converting my mov files to something like Pro-Res? Is there anyway I can do that once they are already cut and in a sequence?

Thank you so much. I get very overwhelmed when unpredictable and strange technicalities occur on a deadline and I appreciate anyone's input!

Jenn

Thank you Shane! Please bear with me here because I am not super technical.

I only have FCP to work with today. So at what point should I convert the clips? If I have already cut mov files into a sequence, can I still convert them all within the sequence? The ideal, since I have already produced my rough cut sequence, would be if I can (in my fantasy dreams it works like this) select the whole clip and click on "convert to....."

I imagine it is not that easy. So...what am I converting it to, how do I do it, and can I do it out of a cut sequence?

Thank you!